I had something similar happen a few years ago. A very old uni friend invited herself to stay with us. She was in a very troubled state, having lots her dad and she was claiming that her DH had been violent to her. So we let her have our empty bedroom.
All well and good. I tried to get help for her, she had run out of her anxiety medication, which she had been prescribed in Spain, where she and her DH had been living. But the medication apparently isn't legal in the UK. That issue was sorted, but when I took her to the chemist, she had a blazing row with her DH, which left me mortified and saying to the chemist, 'I'm going to end up on tranquillisers at this rate.'
For 3 very awkward weeks, she stayed with us. She was obviously not well mentally, she stayed in her dressing gown the whole time, wouldn't let me wash it. She literally reeked.
Then she started playing DH and me against each other, complaining to him that I'd moaned at her about her staying undressed all day. She actually wasn't even decent. She also started cornering my DH, wanting to talk to him on their own.
Long story short, she left, ended up in a psychiatric hospital for a month. She had caused all manner of problems for my friends as well, kept asking them for money. (She owed us a lot of money, I've talked about that on other threads.
So I definitely would say, just don't. This student isn't your responsibility. She needs to go to student welfare.